OFFSET
1,2
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..10000
Alexander Adamchuk's post, Mathematics in Russian, August 29 2006.
Eric Weisstein's World of Mathematics, Catalan Number.
FORMULA
Reciprocal Catalan Constant C = 1 + 4*sqrt(3)*Pi/27.
This number is f(1) where f(x) = -1 + 2*(sqrt(4-x)*(8+x) + 12 * sqrt(x) * arctan(sqrt(x)/sqrt(4-x))) / sqrt((4-x)^5). This form corresponds to a generating function of the reciprocal Catalan numbers in the sense of Sprugnoli. - Juan M. Marquez, Mar 05 2009
Equals -1 + hypergeom([1,2],[1/2],1/4); note hypergeom([1,2],[1/2],x/4) = 1/1 + 1/1*x + 1/2*x^2 + 1/5*x^3 + 1/14*x^4 + 1/42*x^5 + ... is the g.f. for the inverse Catalan numbers (including C(0)). - Joerg Arndt, Apr 06 2013
From Vaclav Kotesovec, May 31 2015: (Start)
Equals 1 + Integral_{x=0..1} Product_{k>=1} (1-x^(9*k))^3 dx.
Equals 1 + Sum_{n>=0} (-1)^n * (2*n+1) / (9*n*(n+1)/2 + 1).
(End)
Equals 1 + Integral_{0..inf} x^3 BesselI_0(x) BesselK_0(x)^2 dx. - Jean-François Alcover, Jun 06 2016
From Amiram Eldar, Jul 05 2020: (Start)
Equals 1 + gamma(4/3)*gamma(5/3).
Equals 1 + Integral_{x=0..oo} dx/(1 + x^3)^2. (End)
EXAMPLE
1.806133050770763489152923670063180325459584999152...
MAPLE
evalf(1 + Sum((-1)^n*(2*n+1)/(9*n*(n+1)/2+1), n=0..infinity), 120); # Vaclav Kotesovec, May 31 2015
MATHEMATICA
RealDigits[N[Sum[n!(n + 1)!/(2n)!, {n, 1, Infinity}], 150]]
RealDigits[N[1+4*Sqrt[3]*Pi/27, 100]][[1]]
PROG
(PARI) default(realprecision, 100); 1 + 4*sqrt(3)*Pi/27
(Magma) SetDefaultRealField(RealField(100)); R:=RealField(); 1 + 4*Sqrt(3)*Pi(R)/27; // G. C. Greubel, Nov 04 2018
CROSSREFS
KEYWORD
cons,nonn
AUTHOR
Alexander Adamchuk, Aug 28 2006
STATUS
approved